> My answer to your question above is similar. The question whether a countable> union of countably many sets is countable is implied by the axiom of countable> choice, which is not provable from ZF. So negating the axiom of countable> choice (which implies negating the axiom of choice) we can have situations> where the conclusion is false.

Feferman and Levy showed that the statement that the set of all realnumbers is the union of a denumerable set of denumerable sets cannotbe refuted.